All models and results in this thesis are based on an empirical demand estimation de-veloped by the former DISPO-team. Parts of this method were already implemented to estimate the demand in terms of the LDP the SLDP and the Price Optimization Problem POP.
Because there are no publications including a detailed description we outline the method at this point.
The overall demandDis considered as an exogenous quantity.1
The estimator circumvents the difficulties arising from varying popularity among different articles by considering the conditional probability that – if an item is sold – this happens in Branchband Sizes. The conditional probability is also called themean amount of demandδb,sfor Branchband Sizes.
Additionally we consider the mean amount of demandδb for Branchb. It is the conditional probability that if an item is sold this happens in Branchb.
1This coincides with the proceeding at our industrial partner. Before the supply according to lots takes place, the purchaser decides on the overall numberDof supplied items.
With these quantities the fractionalmean demanddb,sper branchband sizesfor the setsBof branches andSof sizes is computed. We outline the approach in Subsec-tion3.2.1. We show how to include the setEof scenarios in Subsection3.2.2before we split up db,s to the sales periods – in our case weeks –K\ {kmax}and include price dependency for the price indicesP\ {pmax}in the subsections3.2.3and3.2.4.
In Subsection3.2.5we combine the single estimates to arrive at the estimate we use for DISPO. Concluding, in Subsection3.2.6, we describe how to adapt the demand estimation according to a scenario in effect, as it is the case when we perform price optimization with receding horizon.
Demand is always estimated by pooling observations from products from the same commodity group. A smaller division for example to sub commodity groups in our case would lead to an insufficient amount of data. Commodity groups are for example
“women overgarments classic” or “women overgarments fashion” or “men trousers”.
3.2.1 Relative demand estimation
To exclude the influence of the popularity of the different articles and also to reduce the influence of lost-sales we consider sales just until the day when50%of the observed product’s overall supply (over all branches and sizes) is sold. The advantage of doing so is that we obtain a measurement of the sales speed for different sizes and branches.
If we considered the complete selling time per article varying behavior among branches and sizes – because due to mark-downs nearly all items would be sold out – would not longer be recognizable.
For articlea∈ Awe denote the amount of sales (over all branches and sizes) until the point in time when50%of the overall supply foraare sold bysal50a. The amount of sales for Branchbfor the same time frame is denoted bysal50ab and for Branchb and Sizesbysal50ab,s.
Whenever in this subsection we talk about sales we refer always to the point in time until50%of the supplied items of the considered article are sold.
We define with
δˆba:= sal50ab
sal50a · |B| (3.1)
thescaled relative demandfor branchb for Articlea– scaled in such a way that the mean ofδˆba for Articleaover all branches from the setB takes value one. Because not every observed product may be delivered to all branches this scaling is necessary to guarantee comparability of the observations.
With
δ˜b:=
P
a∈Aˆδba
|A| (3.2)
we define themean relative demandfor Branchbin terms of the set of articlesA.
Themean amount of demandδbfor Branchbis given by the relative frequency δb:=
δ˜b
P
b0∈Bδ˜b0
. (3.3)
Analogously we define the scaled relative demand for Sizesin Branchbfor Article aby
ˆδb,sa :=sal50ab,s
sal50ab · |S| (3.4)
and the mean relative demand for Sizesin Branchbfor the set of articlesAby
˜δb,s :=
P
a∈Aˆδb,sa
|A| . (3.5)
The mean amount of demandδb,sfor Branchband Sizesis given by δb,s:=
δ˜b,s P
s0∈Sδ˜b,s0
. (3.6)
Finally, themean demanddb,sfor Branchband Sizesby
db,s=D·δb·δb,s. (3.7)
That means we split up the estimated overall supplyDto the particular sizes and branches in terms of the observed relative frequencies of sales.
Example 1(relative demand estimation). We want to illustrate the approach on a small example. We assume observations of sales according to the third column of the fol-lowing table. We consider the case of three different observed articles and branches B={b1, b2, b3}. The scaled relative demands per Articleaand Branchbare stated in
In the next table in the second column we stated the mean relative demands per branch in terms of the setAof observed articles. In the third columns the corresponding mean amounts of demand are stated.
b ˜δb δb b1 1.12 0.37 b2 0.90 0.30 b3 0.97 0.33
In the following table in the second column the sales per article for the particular branches and sizess1, s2, s3, s4are stated. The corresponding scaled relative demands are stated in the fourth column.
a b (sal50ab,s
a1 b1 (2,2,2,0) (1.33,1.33,1.33,0.00)
a1 b2 (1,1,0,0) (2.00,2.00,0.00,0.00)
a1 b3 (0,2,2,0) (0.00,2.00,2.00,0.00)
a2 b1 (1,2,0,0) (1.33,2.67,0.00,0.00)
a2 b2 (1,1,1,1) (1.00,1.00,1.00,1.00)
a2 b3 (1,0,1,0) (2.00,0.00,2.00,0.00)
a3 b1 (1,1,0,0) (2.00,2.00,0.00,0.00)
a3 b2 (0,1,1,0) (0.00,2.00,2.00,0.00)
a3 b3 (0,2,1,0) (0.00,2.67,1.33,0.00)
The mean relative demands per branch and size are stated in the next table in the second column. In the third column the mean amount of demand per branch and size is stated.
b (˜δb,s1,δ˜b,s2,˜δb,s3,˜δb,s4) (δb,s1, δb,s2, δb,s3, δb,s4) b1 (1.55,2.00,0.44,0.00) (0.39,0.50,0.11,0.00) b2 (1.00,1.67,1.00,0.33) (0.25,0.42,0.25,0.08) b3 (0.67,1.56,1.78,0.00) (0.17,0.39,0.44,0.00)
On the basis of these computations for a given estimated overall supply ofD= 20we compute the mean demanddb,s per branch and size: For example, the mean demand for Sizes1in Branchb1is given bydb,s= 20·0.37·0.39 = 2.89. The mean demands for all branches and sizes are stated in the following table.
b (db,s1, db,s2, db,s3, db,s4) b1 (2.89,3.70,0.81,0.00) b2 (1.50,2.52,1.50,0.48) b3 (1.12,2.57,2.90,0.00)
3.2.2 Regarding different scenarios
As a next step we include the consideration of different scenarios. The resulting mean demanddeb,sper branchb, sizesand scenarioeis applied in the SLDP, see Section2.3.
Additionally we estimate the scenario probabilitiesProb(e)which are applied in the SLDP and DISPO.
At first we determine the realized scenario for each observed articlea∈ A. We ob-serve sales until two weeks after sales start. For articlea∈ Athe number of these sales over all branches and sizes is given bysal2a and the supply bysup2a. The relation rel2a= sup2sal2aa ∈[0,1]then indicates the popularity of the article. It isrel2a= 0if no item is sold in the first two weeks andrel2a = 1if all items are already sold out after two weeks. We categorize three different scenarios as they are stated in the following table.
rel2a e
<0.33 low seller
≥0.33,≤0.66 normal seller
>0.66 high seller
The scenario probabilitiesProb(e)are given by the relative frequencies of the ob-served scenario in the historical data.
Now we determine how the demand for the low and the high scenario behaves against the normal scenario. This is done the following way:
We categorize the set of articlesAby their scenarios.
For every scenarioe∈ {low seller, normal seller, high seller}thus we obtain a set Ae. The sales until 3 months after sales start for each article over all supplied branches and sizes are given bysala, the supply bysupa. Themean relative salesreleover all articles for one particular scenarioeare given by
rele:=
P
a∈Ae
sala supa
|Ae| . (3.8)
We compute thechange of demanddfefor scenarioeas dfe:= rele
relnormal seller (3.9)
Thus,df = 1for the normal seller scenario.
The mean demand for Scenarioe, Branchband Sizesis given as
deb,s :=dfe·db,s. (3.10)
3.2.3 Splitting up the demand to sales periods
The next step is to split up the demand to the sales periods. This is done by estimating sales rates from the historical data for all periods – in our case weeks. Withsalak we denote the number of sales for Articleaover all branches and sizes in Periodk. We denote withstoakthe overall stock for Articleaat the beginning of Periodk. For Period ktherelative sales per periodrsakfor Articleaare given by
rsak:= salak
stoak. (3.11)
Themean relative sales per periodrsakfor Periodkare given by rsk :=
P
a∈Arsak
|A| . (3.12)
The valuerskequals the mean relative amount of sold items depending on thestock at the beginning of Periodk. We now convert thersk, k = 1, . . . , kmax−1to a factor which describes the amount of sold pieces per size and branch depending on thesupply.
We compute this factor, we call it thesales rate per period srk for Period k, by Algorithm1.
Algorithm 1Sales rates per period
Require: mean relative salesrsk, k∈K\ {kmax} Ensure: sales ratesrk, k∈K\ {kmax}
1: initsum = 0
2: initstock = 1
3: for allk= 0, . . . , kmax−1do
4: nr=rsk·stock
5: sr˜k=nr
6: stock=stock−nr
7: end for
8: for allk∈K\ {kmax}do
9: srk= Psr˜k
j∈Ksr˜k 10: end for
In the for-loop in Step3of Algorithm1the percentage remaining stock (beginning with a stock of one item or100%) is computed according to the mean relative sales.
From the current stock and the mean relative sales the valuessr˜k arises. In Step9of the algorithm the valuessrkare computed by scaling the valuessr˜kin such a way that the sum over all resultingsrk takes value one.
Example 2(sales rates). In this example we assumekmax= 4that means4real sales periods. We assume that the historical data yields mean relative sales as they are stated below.
k 0 1 2 3
rsk 0.5 0.7 0.2 0.4
Now we apply Algorithm1. At firststockis set to value one. In Period0according tors0we sell50%of the current stock or0.5items. The updated stock is set to0.5 andsr˜0 = 0.5. In Period1we start with a stock of0.5. Withrs1 = 0.7, we assume that70%of the current stock that means0.35items are sold. This yields a new stock of0.15. It issr˜1 = 0.35. We proceed analogously for the last two periods and obtain
˜
sr2= 0.03andsr˜3= 0.048.
We scale the valuessr˜k according to Step9of Algorithm1and obtain the results stated in the following table.
k 0 1 2 3
srk 0.539 0.377 0.032 0.052
3.2.4 Price-dependent demand
An important factor in terms of our demand estimation is the influence of the sales price.
To estimate the impact of a mark-down in week k from price πp1 to πp2 with p2> p1the relative sales per week, as defined in the last subsection, for an observed ar-ticleain the week before the mark-downrsak−1and in the week of the mark-downrsak are compared. The observed increase of sales is given by the factorelas˜ aπp
1→πp2 = rsrsaak k−1. Withnoπp1→πp2 beeing the number of articles for which a mark-down fromπp1toπp2
was observed, the mean elasticityelasπp1→πp2 is given by
P
a∈Aelas˜ aπp
1→πp2
noπp1→πp2 .
3.2.5 Combining the estimated factors
The scenario-, time- and price-dependent demand for a product with starting priceπ0 for branchband sizesgiven by
dek,b,s,p =db,s·dfe·srk· Y
p0∈P:p0≤p
elasπ0→πp0. (3.13)
3.2.6 Updating the scenario
In the approach POP-RH, see Subsection2.5.4, we use latest sales figures to determine the scenario in effect. Before we perform POP-RH to adapt our mark-down policy for the subsequent periods we update demand estimation by adapting the factor for the change of demand from Subsection3.2.2. This is done by comparing predicted overall sales with realized overall sales. It issalobsk the amount of sales over all branches and sizes for the last periodk. The amount of predicted sales over all branches and sizes for Periodkis given bysalrealk . Then our updated change of demanddfek is given by dfek =salsalrealobsk
k
.
We compute the dependent demands for the next periodk+ 1by dek+1,b,s,pk =db,s·dfek ·srk+1· Y
p0∈P:p0≤p
elasπp0→πp0. (3.14)